## Abstract

The ontological model framework for an operational theory has generated much interest in recent years. The debate concerning reality of quantum states has been made more precise in this framework. With the introduction of generalized notion of contextuality in this framework, it has been shown that completely mixed state of a qubit is *preparation contextual*. Interestingly, this new idea of preparation contextuality has been used to demonstrate nonlocality of some \(\psi \)-epistemic models without any use of Bell’s inequality. In particular, nonlocality of a non maximally \(\psi \)-epistemic model has been demonstrated from preparation contextuality of a maximally mixed qubit and Schrödinger’s steerability of the maximally entangled state of two qubits (Leifer and Maroney, Phys Rev Lett 110:120401, 2013). In this paper, we, show that any mixed state is preparation contextual. We, then, show that nonlocality of any bipartite pure entangled state, with Schmidt rank two, follows from preparation contextuality and steerability provided we impose certain condition on the epistemicity of the underlying ontological model. More interestingly, if the pure entangled state is of Schmidt rank greater than two, its nonlocality follows without any further condition on the epistemicity. Thus our result establishes a stronger connection between nonlocality and preparation contextuality by revealing nonlocality of any bipartite pure entangled states without any use of Bell-type inequality.

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## Notes

In [14], preparation contextuality has been shown for maximally mixed state of a qubit.

The notion of contextuality is discussed more elaborately in the next section.

This theorem states that every representation of a density matrix \(\rho _B\) can be prepared by acting on a different non-interacting system A if A and B share a pure entangled state \(|\psi _{AB}\rangle \) such that \(\rho _B=Tr_{A} (|\psi _{AB}\rangle \langle \psi _{AB}|)\).

The scenario considered by Leifer and Maroney is more relaxed as it requires to satisfy Eq. (30) only for a single pair of \(\psi \) and \(\phi \).

In [22], Maroney has shown that for every quantum state (in Hilbert spaces of dimension greater than two) there exists another quantum state such that the ontic overlap is strictly less than the quantum overlap and hence the ontological model cannot be maximally \(\psi \)-epistemic.

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## Acknowledgments

It is our great pleasure to acknowledge G. Kar for various discussions, suggestions and help in proving the results. M. B. acknowledges private communications with M. S. Leifer about stronger non-maximal \(\psi \)-epistemicity. Discussion with S. Ghosh is gratefully acknowledged. S. K. C. is thankful to the Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata where a major part of this work was done when he was visiting the unit. A. M. acknowledges support from the CSIR Project 09/093(0148)/2012-EMR-I. S. K. C. acknowledges support from CSIR, Govt. of India. We also like to thank an anonymous referee for useful suggestions.

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Recently, M. S. Leifer have written a review on \(\psi \)-ontology theorems [32]. Among many open questions listed in there one is concerned about preparation contextuality proof for any mixed quantum state. This question is addressed in our paper and we have answered affirmatively.

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Banik, M., Bhattacharya, S.S., Choudhary, S.K. *et al.* Ontological Models, Preparation Contextuality and Nonlocality.
*Found Phys* **44**, 1230–1244 (2014). https://doi.org/10.1007/s10701-014-9839-4

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DOI: https://doi.org/10.1007/s10701-014-9839-4